↑

Properties

Label 2-960-5.4-c3-0-7
Degree $2$
Conductor $960$
Sign $0.0160 + 0.999i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + (−11.1 + 0.178i)5-s + 33.0i·7-s − 9·9-s − 48.3·11-s + 60.3i·13-s + (−0.536 − 33.5i)15-s + 17.7i·17-s − 130.·19-s − 99.2·21-s + 70.8i·23-s + (124. − 4i)25-s − 27i·27-s + 104.·29-s − 210.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.999 + 0.0160i)5-s + 1.78i·7-s − 0.333·9-s − 1.32·11-s + 1.28i·13-s + (−0.00923 − 0.577i)15-s + 0.253i·17-s − 1.58·19-s − 1.03·21-s + 0.642i·23-s + (0.999 − 0.0320i)25-s − 0.192i·27-s + 0.669·29-s − 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0160 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0160 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.0160 + 0.999i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ 0.0160 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4820253411\)
\(L(\frac12)\) \(\approx\) \(0.4820253411\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3iT \)
5 \( 1 + (11.1 - 0.178i)T \)
good7 \( 1 - 33.0iT - 343T^{2} \)
11 \( 1 + 48.3T + 1.33e3T^{2} \)
13 \( 1 - 60.3iT - 2.19e3T^{2} \)
17 \( 1 - 17.7iT - 4.91e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 - 70.8iT - 1.21e4T^{2} \)
29 \( 1 - 104.T + 2.43e4T^{2} \)
31 \( 1 + 210.T + 2.97e4T^{2} \)
37 \( 1 - 300. iT - 5.06e4T^{2} \)
41 \( 1 - 240.T + 6.89e4T^{2} \)
43 \( 1 - 108iT - 7.95e4T^{2} \)
47 \( 1 - 278. iT - 1.03e5T^{2} \)
53 \( 1 + 328. iT - 1.48e5T^{2} \)
59 \( 1 - 889.T + 2.05e5T^{2} \)
61 \( 1 - 241.T + 2.26e5T^{2} \)
67 \( 1 - 103. iT - 3.00e5T^{2} \)
71 \( 1 + 277.T + 3.57e5T^{2} \)
73 \( 1 - 274. iT - 3.89e5T^{2} \)
79 \( 1 + 366.T + 4.93e5T^{2} \)
83 \( 1 + 57.7iT - 5.71e5T^{2} \)
89 \( 1 - 203.T + 7.04e5T^{2} \)
97 \( 1 + 1.28e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.27977306818854730957052407056, −9.282950702873196273193481387472, −8.572540346350207560321510522074, −8.089543048920087776526940778248, −6.84508754555273891502008136429, −5.85190979733750193427746935887, −4.98020717173412635681943499803, −4.15979795246281205144207086746, −2.95578159069212447572066193867, −2.08137921179789906170807221378, 0.18227004539074253846466594989, 0.69979004844029313231538163536, 2.46070596901356179618209060246, 3.60854358608341440167908466835, 4.42111742583327510673412378804, 5.48109088318667325212095964016, 6.76758807636533982879548005677, 7.45324901334651588087071511638, 7.937367687718877388300769067852, 8.684864439190878090450450222531

Graph of the $Z$-function along the critical line